When you write a fraction,
there are actually words for the top number
and the bottom number. And the words are
a lot more fancy than the word "top number"
and "bottom number." What mathematicians
typically use is the word "numerator" for the
top number and "denominator" for the bottom number. And what I want to do now that
we know that the top number is the numerator of the fraction
and the bottom number is the denominator,
I want to compare pairs of fractions that have
either the same denominator or the same numerator. So let's look at
this first pair. I want to compare 4/7 to 3/7. And I have two wholes
right over here. They're the same hole, and I've
divided them into sevenths. I've divided them
into 7 equal chunks. And I want to see what's
larger, 4/7 or 3/7. So what I can do is,
I can fill in 4/7. Let me select 4 out of the 7. So, that's 1, 2, 3, 4. And the fact that trying to even
get to 4/7 I had to have 3/7 first gives you good clue
that 4/7 is probably larger, or it is larger. But now let's color in 3/7,
just so we can compare. So 1, 2, 3/7. And so it's pretty clear
that on the left-hand side, we are shading in
more of the whole than on the right-hand side. So, 4/7 represents
a larger fraction, more of the whole than 3/7. And the way that we can state
that comparison mathematically is with the greater than symbol. We can write 4/7 is
greater than 3/7. Now, the greater than
and less than symbols can sometimes be confusing. This is greater than. This is less than. And the way that
I remember it is that the greater than symbol,
either symbol, the small pointy side is always on the
side of the smaller number, and the big
open side is always on the side of
the larger number. So here, big open
side is opening towards the 4/7, small pointy
side opening to the 3/7. 4/7 is greater than 3/7. Now, what about 3/7 and 3/4? So, here I have
different denominators, but I have the same numerator. And so I encourage you
to pause this video and draw maybe little
boxes like this and try to judge for yourself
which of these is a larger fraction, represents
a larger number. Well, let's color them in. So, let's think about 3/7 first. And we actually
already drew it here, but I can do it really
fast right over here. So that is 3/7. I've colored in 3 of
the 7 equal groups. And what would 3/4 be? Well, that's 1/3, 2/4, and 3/4. And so it's pretty clear
that 3/4 represents a larger fraction of the whole,
that 3/4 is larger, or that 3/7 is smaller. So we could write that
3/7 is less than 3/4. So, notice, same
exact numerator. When I divided it-- because
this fraction symbol could also be
viewed as division. When I have it as
more equal groups, so it's a fraction of
more equal groups, so 3 out of 7 versus 3 out of 4,
this is a smaller number, which which makes sense. Now, let's compare these two. We have the same denominator,
different numerators-- 3/4 versus 2/4. Well, 3/4 we've
already looked at. We can just shade in 3 of these. So 3 of these fourths. So that's 3/4 right over there. And then 2/4, well,
we're going to only have 2 of the fourths, 1, 2. So 2/4 is clearly
the smaller number. 3/4 is the greater number. So, once again, we could
write 3/4 is greater than 2/4. And then finally, I encourage
you to pause the video. Try to come up with whether
2/4 or 3/6 is a larger number. Well, let's color it in again. We've already seen 2/4. We just have to color
in 2 of our fourths. So let's just color in 2 of
our fourths right over there. And then 3/6, we've
split our whole into 6 equal sections--
1, 2, 3, 4, 5, 6. We need to color in 3 of them. And as you see, we're coloring
in the exact same amount of the whole. These two fractions
are equivalent. These are equivalent fractions. 2/4 is equal to 3/6. And as you see
here, they're both filling in half of the whole. If we were to just draw
the whole and split it only-- let me do this
in a different color. If we have our whole, and
we were to split it only into two sections,
we are shading in exactly 1 out
of the 2 sections. So you could say that
2/4 is equal to 3/6, and they both equal 1/2. So 1/2 is equal to
2/4 is equal to 3/6.